Coin-flipping is a deterministic system, so there is no "chance" of landing heads-or-tails, the coin will land in a particular way depending on the initial conditions and the laws of motion.

Because it is a complicated system, and assuming that we are unable to introduce any bias, then modeling the deterministic system with probability gives us what we need to make statistical predictions of the outcome of a (large) number of tosses.

Because probability is being used as a model to provide statistical predictions, you can choose to regard it as modeling either the physics, or your ignorance. The Principal Principle states that we should set our credences to equal the probabilities, so the two very different entities are empirically indistinguishable and often conceptually confused.

Is it a 50% chance for a heads or tails before you flip a coin, or is a 50% chance for a heads or tails after the flip lands but before you look?

*I have my own thoughts I just want to hear what others think. — Jeremiah

According to this, there's a 51% chance of it being whatever was face up when tossed.

If I flip the coin 10 more times each time I flip it, the coin can land on heads or tails, but after it has landed it does not matter how many times I go to look at the coin, it will not change from heads to tails or vice versa.

So what this shows us is that in order for something to have probability there has to be a chance mechanism of some kind involved. After the coin lands probability is no longer a relevant question. We can guess what it might be, and you may have a 50% chance of being right but that chance pertains to your guess and not the coin.

The correct formulation of the problem is that you have a choice to choose heads, tails, or on end or any other outcome you might think of). What actually happens is all full of possibilities due to enumerable forces that surround us. Life is totally unpredictable though some outcomes are more probable than others.

Because probability is being used as a model to provide statistical predictions, you can choose to regard it as modeling either the physics, or your ignorance. The Principal Principle states that we should set our credences to equal the probabilities, so the two very different entities are empirically indistinguishable and often conceptually confused. — tom

In what sense is the model a model of one's ignorance?

Suppose I've never heard of physics and probability theory, and I (incorrectly) expect the outcome of a large number of coin tosses to be influenced by the force of wishful thinking in the vicinity of the coin.

It seems an ordinary probabilistic or statistical model is a model of something in addition to one's ignorance.

We can guess what it might be, and you may have a 50% chance of being right but that chance pertains to your guess — Jeremiah

That's just what we mean when we say that the probability of a coin toss outcome is 50%. So the answer to your question in the OP: it doesn't matter whether the coin toss has occurred or not - as long as you haven't looked.

Suppose I've never heard of physics and probability theory, and I (incorrectly) expect the outcome of a large number of coin tosses to be influenced by the force of wishful thinking in the vicinity of the coin.

It seems an ordinary probabilistic or statistical model is a model of something in addition to one's ignorance. — Cabbage Farmer

Well, if you haven't heard of probability theory (and have no sound intuitions to help you) then all bets are off. We usually assume a "rational agent" who makes the most of the information available to her - otherwise the possibilities are wide open and we cannot say anything at all. But the information available to you doesn't have to include the correct physics. You are right, probability can be a function of a model. But having a model (such as that your wishing can alter the result of a coin toss) influences your expectations - and thus your assignment of probabilities.

So the probabilities that you come up with are still a function of what you believe. If you have no expectations of the behavior of the observed system, then the best you can do is assign ignorance priors (although it should be noted that some don't believe in assigning probabilities in complete ignorance). If you have a model that yields a different result (cheating, power of thought or whatever) then your probabilities will be shaped by the superposition of that model with any remaining uncertainty.

After the coin lands probability is no longer a relevant question. — Jeremiah

I think it is. If we're playing poker we might ask what the probability is that our opponent has a better hand. Simply saying "either he does or he doesn't" isn't a satisfactory answer. A rational person will still play the odds, and not simply toss a coin to determine how to play.

And the fact that the person who still plays the odds is more likely to win shows that probability is still a relevant question after-the-fact. If you have a pair of kings then you bet and if you have an 8 and a 3 then you fold – because your opponent is less likely to already have a better hand than you in the first instance. (For simplicity's sake, imagine we're only playing the cards dealt and not any community cards).

Contrary to @aletheist's opinion on the matter, this doesn't imply anything like Schrödinger cards. The hand your opponent has isn't in some superposition.

Again, my issue is not so much with doing the kinds of calculations that you are describing, it is with calling this "probability" rather than "confidence," "degree of belief," or "plausibility," depending on the context. And I realize that I am swimming upstream here, since people refer to it as "probability" all the time.

The problem I find with those terms (at least with "confidence" and "degree of belief") is that they seem to refer to the strength of someone's (subjective) conviction, but that's not the sort of thing I'm talking about. Probability in the context I'm talking about is an application of actual maths, and has an objective answer. Even after the coin has been flipped, the correct answer to the question "what is the probability that it's heads?" is "0.5", not "very high, as my horoscope tells me that I'm going to be lucky today".

Even after the coin has been flipped, the correct answer to the question "what is the probability that it's heads?" is "0.5" ... — Michael

No, again, at that point the correct answer is 1 if it came up heads and 0 if it came up tails. Someone's personal ignorance of the result is irrelevant once it is actual, rather than potential.

No, again, at that point the correct answer is 1 if it came up heads and 0 if it came up tails. Someone's personal ignorance of the result is irrelevant once it is actual, rather than potential. — aletheist

That's not just how we use the word "probability". We also use the word "probability" to talk about events that have already happened (e.g. the probability that my opponent has a better hand than me). You might prefer the term "confidence" or "degree of belief" or "plausibility" in this situation, but as I explained above, these terms wouldn't be appropriate. When we ordinarily talk about the probability that an already-flipped coin landed heads we're not talking about the strength of our conviction – and nor are we suggesting that the result is in a superposition until we look.

I might have faith that I will win with a pair of sixes, even if I recognise that the odds are not in my favour, and even if I'm aware that my opponent's hand has been dealt and it's either better than mine or it isn't.

I already acknowledged that I am swimming upstream on this. I think that "probability" should be reserved for when we are making predictions about populations and/or random samples, and not applied to events that have already happened such that the only uncertainty about the result is due to ignorance.

Guys, you are both swimming upstream, and for no good reason that I can see.

There are ways of talking about probability in both senses. Probability is always conditionalized on something, if only on the "tautological prior". Let E be an outcome (e.g. heads) of a single coin flip and H be the "fair coin" model, according to which

P(E|H) = 0.5

A different model may yield a different probability.

Now, when you talk about our confidence, we conditionalize on your background knowledge, K:

P(E|K) = ?

If your background knowledge already includes the knowledge of which side the coin came up, the probability will, of course, be either 0 or 1. If you do not know the outcome, and nothing in your background knowledge otherwise informs you about what it might be, and you assume that the coin is fair, then the probability is the same as above:

P(E|H ^ K') = P(E|H) = 0.5

where K' is your background knowledge, not including the proposition "the coin is fair".

Seems to be what I've being saying (albeit more formal)?

Probability is always conditionalized on something ... — SophistiCat

And I am simply suggesting that if it is conditionalized solely on someone's knowledge (or lack thereof), background or otherwise, then we should not call it "probability." Again, I recognize that this is a futile quest.

According to this, there's a 51% chance of it being whatever was face up when tossed. — Michael

The testing protocol was too controlled. What can be said is that under conditions X (where you use a spring mechanized flipping device that offers consistent forces on the coin), the probability of the coin landing on heads is 51% where the coin started out as heads. My assumption is that if the coin flipper were better designed to yield even more consistent and predictable forces, the predictability of the coin landing a particular way would also be affected.

As to the question of what will happen when I flip a coin will only be answered by me flipping a coin, which will be a far less controlled environment. I'll do that and then do a big write up with all sorts of vectors and shit.

Now, when you talk about our confidence, we conditionalize on your background knowledge, K: — SophistiCat

Which is to admit this is all an epistemological question and not an ontological one. That is, you're asking how accurate our knowledge is of an event.

However, if you ask the ontological question of whether the coin is heads or tails prior to looking at it, then you have Schrödinger's cat that is both heads and tails (the coin, not the cat; the cat is dead and alive).

And I am simply suggesting that if it is conditionalized solely on someone's knowledge (or lack thereof), background or otherwise, then we should not call it "probability." Again, I recognize that this is a futile quest. — aletheist

It's not just futile to change a well-established vocabulary, it seems senseless. Is there any particular reason that the word "probability" should not mean what it means?

Which is to admit this is all an epistemological question and not an ontological one. That is, you're asking how accurate our knowledge is of an event. — Hanover

I am not sure what an ontological probability would even mean.

I am not sure what an ontological probability would even mean. — SophistiCat

Note, though, that "ontological probability" is not a term I used. I would refer to ontology as the current state of being, which would describe Schrodinger's cat as both alive and dead at the same time.

That is, if you have a cat in a closed box, and an indeterminate event will determine if cyanide is released inside the box, ontologically the cat is both alive and dead until you open the box.

As Schrödinger says: "The psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts."

Wiki: "When opening the box, the observer becomes entangled with the cat, so "observer states" corresponding to the cat's being alive and dead are formed; each observer state is entangled or linked with the cat so that the "observation of the cat's state" and the "cat's state" correspond with each other."

https://en.wikipedia.org/wiki/Schr%C3%B6dinger's_cat

I don't see what Schrodinger's cat has to do with a coin flip. Of course, probability is involved in both cases, one way or another, but that's not much to go on.

That's just what we mean when we say that the probability of a coin toss outcome is 50%. So the answer to your question in the OP: it doesn't matter whether the coin toss has occurred or not - as long as you haven't looked. — SophistiCat

No, it is not.

Probability and statistics is a useful manmade tool. However, it is not a statement of fact.

If you look at coins, cards or dice, these are all manmade objects designed to be elements for games. The difference between the two sides of a coin, the six sides of a dice, or the difference between cards in a deck, are all based on human subjectivity, and not any objective potential. This is not how nature works.

For example, the hydrogen atom has distinct; quantized, energy levels, each being distinct based on difference in physical potential; energy. The dice is designed to weigh the same on all sides. The difference in the sides of the dice is not a physical potential, but subjective potential, since any decoration, will work the same way. This is not natural reality, but is really based on a type of manmade game reality.

If you designed a dice to mimic the first six energy levels of the hydrogen atom, each side would have different odds depending on the situation. Each side would weigh differently, based on its potential. This is not how dice work, since they are manmade. The sides of a dice alway weigh the same in terms of potential. The difference is superficial and subjective.

Say we take a deck of cards. The cards in the deck only differ by the picture that is printed on the face of the card. This print has no physical potential. If you burn a 10 and a 2, the 10 does not give off more energy, like in nature. The potential within the cards is imaginary; game. Say I erase all the pictures, how does this affect the odds? The odds become 1.0 that a blank card will be drawn. The subjective element controls the odds.

Statistics has to be one of the best magic tricks in science, with almost everyone fooled. It is a tool whose game assumptions have been turned into the facts of nature, and nobody sees the subjective pitfalls within the game itself.

Probability and statistics is a useful manmade tool. However, it is not a statement of fact. — wellwisher

No one here is claiming statistics are facts and in no respectable statistics text book is anyone rightfully claiming statistics are facts. In fact, they teach just the opposite. Everyone here knows statistics are estimating unknown parameters. So, you can put the soapbox away.

Also, Nature made humans and humans do what is in their nature to do; your division between what is natural and what is "man-made" is subjective and objectively does not exist. And odds are a ratio, it's chance that runs 0 to 1.

Humans can come up with a number of theories for the same thing; creation of the universe. Nature does not work that way. The creation of the universe happened one way even of we are not sure what that one way was.

The laws of physics are the same in all references, yet different theories use different mental references to explain the same things; string theory versus standard theory. These cannot both be true. They are mutually exclusive; the subjective faces on the dice.

Nature weighs each side of its dice, differently by using universal potentials; loaded dice. Manmade theory changes the faces of the cards, to suit the theory; n-dimensions or 4-D. The nebulous nature of statistics, allows this misdirect to stand, since it is not designed to address the conceptual foundations on the card faces. It accepts the card faces as givens.

If we took away statistics, this would throw sand in the gears. We would need to address the face of the cards. But then again, as you said, humans are designed to extrapolate nature; manmade, with manmade benefitting by statistics. My problem with statistics is not connected to manmade, but to natural, since statistics allows manmade card faces to infiltrate natural, until it appears to be real based on odds.

For example, we can make diamonds in the lab using an hot anvil press at 1 million PSI. This can take hours instead of millions of years as is assumed for natural diamonds. Hours is manmade and millions years is natural.

Say I decide nature to present an anvil press theory for natural diamonds, to prove that natural diamonds are also created in hours. Statistics can be used to make this appear real. I can run the same lab tests for both natural and manmade.

If we get rid of statistics for natural things, we need to go back to the premises, and examine these, since not all face of cards use natural potentials. For manmade, statistics can be quite useful since the goal is a marketable product and not its detailed science. People are allowed to gamble if they chose, But God did not play dice with the natural universe; not statistical.

Man who upon only seeing 12 heads in a row still assumes fair coin is silly man. Man who assumes it's a double header upon only seeing 12 heads is almost as silly.

Man who assumes it's a double header upon only seeing 12 heads is almost as silly. — fdrake

What about 20? That's 1 in 9.5e-7, which is more than 3 times as rare as the 5-sigma confidence used in physics.

Yeah. But most coins are not loaded. And many coins are flipped!

I don't think you actually know anything about statistics. I think you are confusing probability, a branch of mathematics with the branch of science known as statistics. Which, actually is an understandable mistake; however, the conceptual aspects of statistics is largely contained in the science.

Actually all coins are loaded, no coin flip is exactly 1:1. Fair coin flips only exist in our imagination or possibly also computer simulations, I don't know computers well enough to say. Otherwise we are working on a good enough to be called fair basis, but then that is nearly all of mathematics.

The truth is all of math and all of science is a lie based on inaccurate assumptions, but they are still the best tools we have.

C'mon man, this is just pedantic. Fair's the best model a priori as it's a pretty good approximation to the base model odds of heads vs tails close to 1. You need relatively a lot of data (long sequences of flips) to distinguish a generic coin from a perfectly fair one, even though irregularities will probably make any given coin slightly non-fair. This is 'cos fair coins produce maximally variant flips.

You can't even claim 'the coin is unfair' from typical short sequences of flips that are how coin flips are used.

The truth, here, is almost useless.